Conservation Laws and Boundary Conditions

Wave and Wave Body Interactions
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We begin by deriving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated Navier Stokes equations. However, for the large scale processes that occur in ocean waves many simplifications are possible.

Coordinate system and velocity potential

Coordinate System

We begin by defining the coordinate system.

$\begin{matrix} &(x,y,z) &: &\mbox{Coordinate system} \\ &\mathbf{x} &: &\mbox{Fixed Eulerian Vector} \\ & \mathbf{v} &: &\mbox{Flow Velocity Vector at} \ \mathbf{x} \\ &\zeta &: &\mbox{Free Surface Elevation} \\ &\mathbf{g} &: &\mbox{Acceleration due to gravity} \end{matrix}$

At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.

The most important assumption we make is that the fluid is an ideal fluid, i.e. there are no shear stresses due to viscosity and that the flow is irrotational. This means that

$\nabla \times \mathbf{v} = 0$

We now introduce the very important concept of the velocity potential. Essentially this allows us to express the solution as a function of a scalar (a function which has a single value as opposed to a vector function which has multiple values) rather than a vector throughout the fluid domain. There is an important theorem in vector calculus [1] that if $\nabla \times \mathbf{v} = 0$ then we can express the irrotational vector as the gradient of a scalar function, i.e.

$\mathbf{v} = \nabla \Phi$

where $\Phi(\mathbf{x},t)$ is called the velocity potential.

It turns out that the potential flow model of surface wave propagation and wave-body interactions is very accurate for the kind of length and time scales which are important in the ocean. It is important to realise, however, that we have made considerable simplifications and that certain processes, most notably wave breaking, are in no way covered by this theory. In fact, the process of wave breaking is extremely complicated and is much less well understood than the potential flow model.

Conservation of mass

The key equation we will solve to understand ocean waves is Laplace's equation which is derived very simply from the expression for the velocity potential. We assume that the fluid is incompressible so that conservation of mass gives us the following condition

$\nabla \cdot \mathbf{v} = 0$

This condition in turn implies, using the definition of the velocity potential that

$\nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0$

or

$\partial_x^2 \Phi + \partial_y^2\Phi + \partial_z^2\Phi = 0,$

which is Laplace's equation.

Conservation of linear momentum

We begin with Euler's equation in the absence of viscosity

$\partial_t \mathbf{v} + (\mathbf{v}\cdot \nabla)\mathbf{v}= - \frac1{\rho} \nabla P + \mathbf{g}$

where $P(\mathbf{x}, t)$ is the fluid Pressure at $(\mathbf{x}, t)$ and $\mathbf{g}= - \mathbf{k} g$ is the acceleration due to gravity where $\mathbf{k}$ is the unit vector pointing in the positive $z$-direction (so we are now setting the $z$ coordinate to point in the vertical direction). Finally $\rho$ is the water density.

We then use the following vector identity

$(\mathbf{v} \cdot \nabla) \mathbf{v} = \frac 1{2} \nabla (\mathbf{v} \cdot \mathbf{v}) - \mathbf{v}\times ( \nabla \times \mathbf{v})$

and since we have irrotational flow (i.e. $\nabla \times \mathbf{v}= 0$) Euler's equation becomes

$\partial_t \mathbf{v} + \frac{1}{2} \nabla (\mathbf{v} \cdot \mathbf{v}) = - \frac 1{\rho} \nabla P - \nabla (g z)$

where we have used $\nabla z = \mathbf{k}$. We now substitute $\mathbf{v}= \nabla \Phi$ and we obtain

$\nabla (\partial_t \Phi + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g z ) = 0.$

We now observe that if

$\nabla F( \mathbf{x}, t) =0 \quad \Longrightarrow \quad F (\mathbf{x}, t) = C$

where $C$ is an arbitrary constant.

Bernoulli's equation

Bernoulli's equation follows from the equation above.

$\partial_t \Phi + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g z = C$

or

$\frac{P}{\rho} = - \partial_t\Phi-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g z + C$

The value of the constant $C$ is immaterial (it can be thought of as defining the reference pressure. It is also worth noting that the angular momentum conservation principle is contained in $\nabla \times \mathbf{v} = 0.$ In particular, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

Derivation of nonlinear free-surface condition

A very important result is the boundary condition at the free surface of the fluid and air. There are two conditions which relate the free surface displacement $\zeta(x,y,t)$ and the velocity potential $\Phi(x,y,z,t)$ at the free surface. The dynamic condition is derived from the Bernoulli's equation and the kinematic condition is derived from the equations linking the fact that the velocity vector at the surface is given by the gradient of the potential. We will present two methods to derive these equations.

Method I

We derive the dynamic condition directly from Bernoulli's equation. On $z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure}$. This allows us to rewrite Bernoulli's equation as

$\frac{P_a}{\rho}=-\frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi\cdot\nabla\Phi-g\zeta+C \qquad \mbox{on} \ z=\zeta(x,y,t)$

We will simplify this equation by showing that we are free to set the pressure to any value.

The kinematic condition is derived as follows. On $z=\zeta$ The mathematical function

$z-\zeta(x,y,t)\equiv\tilde{f}(x,y,z,t)$

is always zero when tracing a fluid particle on the free surface. So the substantial or total derivative of $\tilde{f}$ must vanish, thus

$\frac{D\tilde{f}}{Dt}=0=\left (\partial_t + \mathbf{v} \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ z=\zeta$

Expanding, we obtain

$\left (\partial_t + \mathbf{v} \cdot \nabla \right ) (z-\zeta) =0, \qquad \mbox{on} \ z=\zeta$

which in turn implies that

$\partial_t\zeta + \partial_x\Phi \partial_x\zeta + \partial_y\Phi \partial_y\zeta = \partial_z\Phi, \qquad \mbox{on} \ z=\zeta$

which is the Kinematic free-surface condition.

We have already derived the dynamic condtion from Bernoulli's equation

$\partial_t\Phi + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = C - \frac{P_a}{\rho}, \qquad \mbox{on} \ z=\zeta$

Constants in Bernoulli's equation may be set equal to zero when we are eventually interested in integrating pressures over closed or open boundaries (floating or submerged bodies) to obtain forces & moments. This follows from a simple application of one of the two Gauss vector theorems.

Gauss theorem

Force coordinates

We need to use the following theorems often called Gauss theorem although more properly known as the divergence theorem. We begin with the vector version. If $\mathbf{n}$ is the unit normal vector pointing inside the volume $V$ with surface $S$ and $f(\mathbf{x})$ is an arbitrary sufficiently differentiable scalar function, then

$\iiint_{\Omega} \nabla f \mathrm{d}v = -\iint_{\partial\Omega} f_{s} \mathbf{n} \mathrm{d}s$

Note the three scalar identities that follow:

$\begin{align} \iiint_{{\Omega}} \partial_x f \mathrm{d}v &= - \iint_{\partial\Omega} f n_1 \mathrm{d}s \\ \iiint_{{\Omega}} \partial_y f \mathrm{d}v &= - \iint_{\partial\Omega} f n_2 \mathrm{d}s \\ \iiint_{{\Omega}} \partial_z f \mathrm{d}v &= - \iint_{\partial\Omega} f n_3 \mathrm{d}s . \end{align}$

The scalar version is as follows where $\mathbf{v}$ is an arbitrary sufficiently differentiable vector function

$\iiint_{\Omega} \nabla \cdot \mathbf{v} = - \iint_{\partial\Omega} \mathbf{v} \cdot \mathbf{n} \mathrm{d}s$

The scalar identity is often used to prove mass conservation principle.

Definition of force and moment in terms of fluid pressure

The force $\mathbf{F}$ is given by

$\mathbf{F} = \iint_{\partial\Omega} P\mathbf{n}\mathrm{d}s$

where $P$ is the pressure and the moment $\mathbf{M}$ is given by

$\mathbf{M} = \iint_{\partial\Omega} P(\mathbf{x}\times\mathbf{n})\mathrm{d}s$

It follows from the Gauss theorem that if $P = C$ the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set $C=0$. It follows that the dynamic free surface condition takes the form

$\zeta (x,y,t) = - \frac{1}{g} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad z=\zeta$

Method II

When tracing a fluid particle on the free surface the hydrodynamic pressure given by Bernoulli (after the constant $C$ has been set equal to zero) must vanish as we follow the particle:

$\frac{D}{Dt} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi+gz \right \} =0, \qquad z=\zeta$

or

$\left ( \partial_t + \mathbf{v} \cdot \nabla \right ) \left ( \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi +gz \right ) =0, \qquad z=\zeta$

This condition also follows upon elimination of $\zeta$ from the kinematic & dynamic conditions derived under method I.

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This article is based on the MIT open course notes and the original article can be found here